Various procedures for balancing successive rotary members are carried out in automatically balancing machines, involving effecting an unbalance measuring operation in one or more planes on each rotary member which rotates during that operation at a given speed of rotation for measurement purposes, within a given unbalance measuring period in which the required items of measuring information are ascertained in a plurality of repetitive measuring cycles. Operating steps are then carried out on the rotary member to provide for balancing thereof. Automatically operating balancing machines of that kind are frequently associated with transportation assemblies or transfer tools with which the respective rotary members to be measured are introduced into the measuring station of the balancing machine and removed again therefrom and then transported away for further operations. As indicated above, a suitable balancing operation can be carried out on the rotary member after the conclusion of a measuring run and an unbalance measuring operation, and that may involve removing material from the rotary member by a suitable machining operation such as drilling or milling, or balancing weights may be secured to the rotary member. That balancing operation can be carried out in the measuring station itself or in a separate balancing station, still within the automatic balancing machine. However those working operations are inevitably such as to give rise to vibration which can affect the unbalance measuring operation in such a way as to falsify the measurement results thereof. The temporal succession of the individual working steps involved in that procedure such as transportation of the rotary member, measuring the rotary member unbalance and effecting the balancing operation is therefore controlled in automatic balancing machines in many cases in such a way that, during the unbalance measuring operation, there are no other movements taking place within the assembly, which would give rise to vibration. As the unbalance measuring operation takes from four to ten seconds, depending on the particular type of rotary member, in order to ascertain the required number of items of measuring information, it is difficult to provide a procedure, which is advantageous in respect of time, in regard to the individual maching steps which are to be carried out on the rotary member, in an automatically operating balancing system.